Functions:Injective

A function $f:A\to B$ is injective, or "one-to-one", if for all $a_{1},a_{2}\in A$, $a_{1}\neq a_{2}$ implies that $f(a_{1})\neq f(a_{2})$ (or for all $a_{1},a_{2}\in A$, $f(a_{1})=f(a_{2})$ implies that $a_{1}=a_{2}$). That is, a function is injective if each output is unique to a specific input, and no two distinct inputs map to the same output.

Examples:

• Let $A=\{1,2,3\}$ and $B=\{1,2,3\}$, and let $f:A\to B$ such that $f(1)=2$, $f(2)=1$, and $f(3)=3$. $f$ is an injective function because each output of $f$ is mapped to by exactly one input.

• Let $g:A\to B$ such that $g(1)=1$, $g(2)=1$, and $g(3)=2$. $g$ is not an injective function since $g(1)=g(2)$.

• $f:\mathbb {R} \to \mathbb {R}$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x } is an injective function, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1 \neq x_2 \implies f(x_1) \neq f(x_2) } for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1,x_2\in\R } .

• Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:\R\to\R } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^2 } . This function is NOT injective because for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \neq 0 } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^2 = (-x)^2 = f(-x) } , but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \neq -x } . For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\neq -1} while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(1) = f(-1) = 1} , which conflicts with the definition of injectivity.

Resources

• Injective Function, Wikipedia
• Function Types, OpenStax
• Proofs and Fundamentals: A First Course in Abstract Mathematics, pages 154-164

Also see functions.